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Magnetostatic modes and magnetic polaritons in ferromagnetic and antiferromagnetic nanorings
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 2183–2185 www.elsevier.com/locate/jmmm
Magnetostatic modes and magnetic polaritons in ferromagnetic and antiferromagnetic nanorings T.K. Das, M.G. Cottam Department of Physics and Astronomy, University of Western Ontario, London, Ont., N6A 3K7 Canada Available online 15 November 2006
Abstract Magnetostatic mode calculations are presented for the bulk and surface modes in long cylindrical magnetic nanorings with arbitrary values of the inner and outer radii. Numerical examples for the dispersion relations are presented for the cases of the ferromagnet Ni and the uniaxial antiferromagnet GdAlO3. The generalization to include retardation effects and hence magnetic polaritons in a ring is briefly indicated. r 2006 Elsevier B.V. All rights reserved. PACS: 76.30.D; 41.20.G Keywords: Magnetostatic mode; Nanoring; Ferromagnet; Antiferromagnet; Polariton
Recently, the fabrication of high-density arrays of magnetic nanorings, consisting of materials such as permalloy or nickel, was reported [1], as well as experimental studies of their spin dynamics by Brillouin light scattering (BLS) and magnetic resonance (see Ref. [2]). This has motivated us to present an analytic theory for the longwavelength magnetostatic modes. It applies to ferromagnetic and antiferromagnetic materials with the external magnetic field and static magnetization taken parallel to the cylinder axis of a nanoring, which is assumed to have a large length-to-diameter aspect ratio. Both surface and bulk modes of a nanoring are considered, and applications are made to different materials. The inclusion of retardation effects to obtain magnetic polaritons is briefly discussed. The nanorings are modeled as long hollow cylinders with inner and outer radii R1 and R2, respectively. A nonmagnetic medium fills the regions ro R1 and r4R2. By contrast with Ref. [3], where a theory of nanorings with negligible wall thickness (R1ER2) was developed, we allow these parameters to take general values. We thus incorporate the special cases of a wire (R1-0) and an ‘‘antiwire’’ (R2-N) consisting of a nonmagnetic cylinder embedded in a magnetic medium. In Corresponding author. Tel.: +1 519 661 2111; fax: +1 519 661 3703.
E-mail address: [email protected] (M.G. Cottam). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.752
considering the magnetostatic modes we are focusing on situations where the long-range dipole–dipole interactions dominate in the spin dynamics, compared to the exchange. This is the case provided the wave number q along the symmetry axis of the nanoring is sufficiently small (typically 107 m1 or less), which is achievable in BLS if a 901 scattering geometry is employed. By contrast, the BLS experiments in Ref. [2] involved a 1801 backscattering geometry giving a larger q corresponding to the dipole-exchange region. Typical outer radii are of order 50 nm. As in the wire case, which was analyzed previously [4], we employ the nondiagonal magnetic susceptibility tensor with the frequency-dependent components wxx ¼ wyywa and wxy ¼ wyxiwb. This is the form when the z-axis, corresponding to the direction of the applied field B0 and the saturation magnetization M0, coincides with the nanoring axis. For a ferromagnet the expressions [4,5] are wa ¼ (o0/o)wb ¼ o0om/(o20o2) with o denoting the angular frequency of the wave, while o0 ¼ gB0 and om ¼ gm0M0 (g ¼ gyromagnetic ratio). For a uniaxial antiferromagnet wa and wb have analogous expressions [5,6] but additional parameters oa and oex (related to the anisotropy and static exchange, respectively) are involved. The poles for o occur at O07o0, where O0 ¼ [oa (oa+2oex)]1/2 is the antiferromagnetic resonance angular frequency [5].
ARTICLE IN PRESS T.K. Das, M.G. Cottam / Journal of Magnetism and Magnetic Materials 310 (2007) 2183–2185
2184
Briefly, the calculation in the magnetostatic case proceeds by generalizing previous work for ferromagnets in the limiting case of a wire [4]. The appropriate Maxwell’s equations are re-expressed in terms of the magnetostatic scalar potential c, which satisfies the Walker equation inside the magnetic material and Laplace’s equation outside. For the ring case the solutions have the form fm,q (r) exp(imy+iqz) in cylindrical polar coordinates (m is an integer and q is the wave number). The solutions for the radial function involve the Bessel functions Im (qr) for ro R1, Im (aqr) and Km(aqr) for R1o ro R2, and Km(qr) for r4R2, where a is a o-dependent parameter defined by a2 ¼ (1+wa)1. It is real or imaginary for surface modes (localized near the interfaces) or bulk modes (with a wavelike behavior across the ring thickness), respectively. On applying the magnetostatic boundary conditions at r ¼ R1 and R2, we find the dispersion relation for the bulk and surface modes as I m ðqR1 Þ c1 d 1 0 0 0 I m ðqR1 Þ c1 FK1 d 1 FI1 ¼ 0, 0 c2 d2 K m ðqR2 Þ 0 c2 FK2 d 2 FI2 K 0m ðqR2 Þ where FKi ¼ [K0 m(aqRi)/aKm(aqRi)][mwb/qRi] with a similar definition for FIi ; also ci ¼ Km(aqRi), and di ¼ Im(aqRi), (i ¼ 1 or 2). The above form of the result is applicable for both ferromagnetic and antiferromagnetic materials. In Fig. 1 we show calculations for nickel nanorings taking o0/2p ¼ 15.5 GHz (B0 ¼ 0.5 T) and om/2p ¼ 18.7 GHz, with frequencies plotted versus qR2 for |m| ¼ 1 and 2 (as labeled) for two values of R1/R2. The surface modes occur for o0 þ 12om 4o4½o0 ðo0 þ om Þ1=2 , where the limits are shown as horizontal lines. The maximum frequency is always at q ¼ 0 and varies with both |m| and R1/R2. For each |m| there are two modes (since there are two interfaces), existing only for q less than a threshold
17.9
Ni B0= 0.5 T
2 24.5
17.8
R1/R2= 0.9 ω/2π (GHz)
24.0
23.5
GdAlO3
2
R1/R2= 0.4
1 ω/2π (GHz)
value (as for a wire [4]). The quantized bulk modes have frequencies below the surface band and generally can occur for larger qR2 values than the surface modes. For example, at qR2 ¼ 10 and R1/R2 ¼ 0.4 the lowest two modes, i.e. those with zero and one nodes in fm,q (r) for R1o ro R2, occur at 17.5 and 18.2 GHz for |m| ¼ 1. The frequencies generally increase with the number of nodes. Our theory becomes consistent with [3,4] in the appropriate limiting cases. In Fig. 2 we show some results for the antiferromagnet GdAlO3 with o0/2p ¼ 19.6, om/2p ¼ 22.0, oa/2p ¼ 10.2, and oex/2p ¼ 52.6 (all in GHz). The relatively low AFMR frequency corresponds to O0/2p ¼ 34.4 GHz. In the antiferromagnetic case the surface modes exist only when B06¼0 and occur within a narrow band just above O0o0 as indicated by the horizontal lines. In other respects the surface-mode properties are qualitatively similar to those for ferromagnetic rings, but the quantized bulk modes now occur in two bands above and below the surface modes. Finally, we have extended some of these calculations using the full form of Maxwell’s equations to describe magnetic polaritons, following a procedure established in the context of electromagnetic wave propagation in ferrite rods [7]. It is complicated because the TE and TM modes cannot be separated. Consequently, additional Bessel function (with modified propagation wave numbers) appear in the general solution. Applying the electromagnetic boundary conditions, an 8 8 determinantal condition is found, reducing to 4 4 in the wire and antiwire cases. As usual [6,7], the modes most affected by retardation are those near the ‘‘light line’’ in the dispersion curve. In a ring geometry this means that if qR2 has a magnitude much larger than that of order noresR2/c (where n is the refractive index, c is the speed of light, and ores denotes o0 or O0 for a ferromagnet or antiferromagnet, respectively) then retardation effects are very small. For Ni or GdAlO3 rings in BLS studies this implies minimal
B0= 0.7 T
1
R1/R2= 0.5 R1/R2= 0.8
17.7 2 17.6
2
1 17.5
1 23.0 0
1
2
3
4
5
qR2
17.4 0
1
2
3
4
qR2 Fig. 1. Frequencies of surface magnetostatic modes in nickel nanorings versus wave number q. See text for details.
Fig. 2. Same as Fig. 1 but for antiferromagnetic GdAlO3.
5
ARTICLE IN PRESS T.K. Das, M.G. Cottam / Journal of Magnetism and Magnetic Materials 310 (2007) 2183–2185
retardation if R2 ¼ 1 mm or less but significant retardation if, e.g., R21 mm. We are thankful for the helpful discussions with T.M. Nguyen. References [1] K.L. Hobbs, P. Larson, G. Lian, J. Keay, M. Johnson, Nano-Lett. 4 (2004) 167.
2185
[2] Z.K. Wang, H. Lim, H. Liu, S. Ng, M. Kuok, L. Tay, D. Lock wood, M. Cottam, K. Hobbs, P. Larson, J. Keay, G. Lian, M. Johnson, Phys. Rev. Lett. 94 (2005) 137208. [3] H. Leblond, V. Veerakumar, Phys. Rev. B 70 (2004) 134413. [4] T.M. Sharon, A. Maradudin, J. Phys. Chem. Solids 38 (1977) 977. [5] T. Wolfram, R. De Wames, Prog. Surf. Sci. 2 (1972) 233. [6] N.S. Almeida, G. Farias, N. Oliveira, E. Vasconcelos, Phys. Rev. B 48 (1993) 9839. [7] F. Schott, T. Tao, R. Freibun, J. Appl. Phys. 38 (1967) 3015.
Journal of Magnetism and Magnetic Materials 310 (2007) 2183–2185 www.elsevier.com/locate/jmmm
Magnetostatic modes and magnetic polaritons in ferromagnetic and antiferromagnetic nanorings T.K. Das, M.G. Cottam Department of Physics and Astronomy, University of Western Ontario, London, Ont., N6A 3K7 Canada Available online 15 November 2006
Abstract Magnetostatic mode calculations are presented for the bulk and surface modes in long cylindrical magnetic nanorings with arbitrary values of the inner and outer radii. Numerical examples for the dispersion relations are presented for the cases of the ferromagnet Ni and the uniaxial antiferromagnet GdAlO3. The generalization to include retardation effects and hence magnetic polaritons in a ring is briefly indicated. r 2006 Elsevier B.V. All rights reserved. PACS: 76.30.D; 41.20.G Keywords: Magnetostatic mode; Nanoring; Ferromagnet; Antiferromagnet; Polariton
Recently, the fabrication of high-density arrays of magnetic nanorings, consisting of materials such as permalloy or nickel, was reported [1], as well as experimental studies of their spin dynamics by Brillouin light scattering (BLS) and magnetic resonance (see Ref. [2]). This has motivated us to present an analytic theory for the longwavelength magnetostatic modes. It applies to ferromagnetic and antiferromagnetic materials with the external magnetic field and static magnetization taken parallel to the cylinder axis of a nanoring, which is assumed to have a large length-to-diameter aspect ratio. Both surface and bulk modes of a nanoring are considered, and applications are made to different materials. The inclusion of retardation effects to obtain magnetic polaritons is briefly discussed. The nanorings are modeled as long hollow cylinders with inner and outer radii R1 and R2, respectively. A nonmagnetic medium fills the regions ro R1 and r4R2. By contrast with Ref. [3], where a theory of nanorings with negligible wall thickness (R1ER2) was developed, we allow these parameters to take general values. We thus incorporate the special cases of a wire (R1-0) and an ‘‘antiwire’’ (R2-N) consisting of a nonmagnetic cylinder embedded in a magnetic medium. In Corresponding author. Tel.: +1 519 661 2111; fax: +1 519 661 3703.
E-mail address: [email protected] (M.G. Cottam). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.752
considering the magnetostatic modes we are focusing on situations where the long-range dipole–dipole interactions dominate in the spin dynamics, compared to the exchange. This is the case provided the wave number q along the symmetry axis of the nanoring is sufficiently small (typically 107 m1 or less), which is achievable in BLS if a 901 scattering geometry is employed. By contrast, the BLS experiments in Ref. [2] involved a 1801 backscattering geometry giving a larger q corresponding to the dipole-exchange region. Typical outer radii are of order 50 nm. As in the wire case, which was analyzed previously [4], we employ the nondiagonal magnetic susceptibility tensor with the frequency-dependent components wxx ¼ wyywa and wxy ¼ wyxiwb. This is the form when the z-axis, corresponding to the direction of the applied field B0 and the saturation magnetization M0, coincides with the nanoring axis. For a ferromagnet the expressions [4,5] are wa ¼ (o0/o)wb ¼ o0om/(o20o2) with o denoting the angular frequency of the wave, while o0 ¼ gB0 and om ¼ gm0M0 (g ¼ gyromagnetic ratio). For a uniaxial antiferromagnet wa and wb have analogous expressions [5,6] but additional parameters oa and oex (related to the anisotropy and static exchange, respectively) are involved. The poles for o occur at O07o0, where O0 ¼ [oa (oa+2oex)]1/2 is the antiferromagnetic resonance angular frequency [5].
ARTICLE IN PRESS T.K. Das, M.G. Cottam / Journal of Magnetism and Magnetic Materials 310 (2007) 2183–2185
2184
Briefly, the calculation in the magnetostatic case proceeds by generalizing previous work for ferromagnets in the limiting case of a wire [4]. The appropriate Maxwell’s equations are re-expressed in terms of the magnetostatic scalar potential c, which satisfies the Walker equation inside the magnetic material and Laplace’s equation outside. For the ring case the solutions have the form fm,q (r) exp(imy+iqz) in cylindrical polar coordinates (m is an integer and q is the wave number). The solutions for the radial function involve the Bessel functions Im (qr) for ro R1, Im (aqr) and Km(aqr) for R1o ro R2, and Km(qr) for r4R2, where a is a o-dependent parameter defined by a2 ¼ (1+wa)1. It is real or imaginary for surface modes (localized near the interfaces) or bulk modes (with a wavelike behavior across the ring thickness), respectively. On applying the magnetostatic boundary conditions at r ¼ R1 and R2, we find the dispersion relation for the bulk and surface modes as I m ðqR1 Þ c1 d 1 0 0 0 I m ðqR1 Þ c1 FK1 d 1 FI1 ¼ 0, 0 c2 d2 K m ðqR2 Þ 0 c2 FK2 d 2 FI2 K 0m ðqR2 Þ where FKi ¼ [K0 m(aqRi)/aKm(aqRi)][mwb/qRi] with a similar definition for FIi ; also ci ¼ Km(aqRi), and di ¼ Im(aqRi), (i ¼ 1 or 2). The above form of the result is applicable for both ferromagnetic and antiferromagnetic materials. In Fig. 1 we show calculations for nickel nanorings taking o0/2p ¼ 15.5 GHz (B0 ¼ 0.5 T) and om/2p ¼ 18.7 GHz, with frequencies plotted versus qR2 for |m| ¼ 1 and 2 (as labeled) for two values of R1/R2. The surface modes occur for o0 þ 12om 4o4½o0 ðo0 þ om Þ1=2 , where the limits are shown as horizontal lines. The maximum frequency is always at q ¼ 0 and varies with both |m| and R1/R2. For each |m| there are two modes (since there are two interfaces), existing only for q less than a threshold
17.9
Ni B0= 0.5 T
2 24.5
17.8
R1/R2= 0.9 ω/2π (GHz)
24.0
23.5
GdAlO3
2
R1/R2= 0.4
1 ω/2π (GHz)
value (as for a wire [4]). The quantized bulk modes have frequencies below the surface band and generally can occur for larger qR2 values than the surface modes. For example, at qR2 ¼ 10 and R1/R2 ¼ 0.4 the lowest two modes, i.e. those with zero and one nodes in fm,q (r) for R1o ro R2, occur at 17.5 and 18.2 GHz for |m| ¼ 1. The frequencies generally increase with the number of nodes. Our theory becomes consistent with [3,4] in the appropriate limiting cases. In Fig. 2 we show some results for the antiferromagnet GdAlO3 with o0/2p ¼ 19.6, om/2p ¼ 22.0, oa/2p ¼ 10.2, and oex/2p ¼ 52.6 (all in GHz). The relatively low AFMR frequency corresponds to O0/2p ¼ 34.4 GHz. In the antiferromagnetic case the surface modes exist only when B06¼0 and occur within a narrow band just above O0o0 as indicated by the horizontal lines. In other respects the surface-mode properties are qualitatively similar to those for ferromagnetic rings, but the quantized bulk modes now occur in two bands above and below the surface modes. Finally, we have extended some of these calculations using the full form of Maxwell’s equations to describe magnetic polaritons, following a procedure established in the context of electromagnetic wave propagation in ferrite rods [7]. It is complicated because the TE and TM modes cannot be separated. Consequently, additional Bessel function (with modified propagation wave numbers) appear in the general solution. Applying the electromagnetic boundary conditions, an 8 8 determinantal condition is found, reducing to 4 4 in the wire and antiwire cases. As usual [6,7], the modes most affected by retardation are those near the ‘‘light line’’ in the dispersion curve. In a ring geometry this means that if qR2 has a magnitude much larger than that of order noresR2/c (where n is the refractive index, c is the speed of light, and ores denotes o0 or O0 for a ferromagnet or antiferromagnet, respectively) then retardation effects are very small. For Ni or GdAlO3 rings in BLS studies this implies minimal
B0= 0.7 T
1
R1/R2= 0.5 R1/R2= 0.8
17.7 2 17.6
2
1 17.5
1 23.0 0
1
2
3
4
5
qR2
17.4 0
1
2
3
4
qR2 Fig. 1. Frequencies of surface magnetostatic modes in nickel nanorings versus wave number q. See text for details.
Fig. 2. Same as Fig. 1 but for antiferromagnetic GdAlO3.
5
ARTICLE IN PRESS T.K. Das, M.G. Cottam / Journal of Magnetism and Magnetic Materials 310 (2007) 2183–2185
retardation if R2 ¼ 1 mm or less but significant retardation if, e.g., R21 mm. We are thankful for the helpful discussions with T.M. Nguyen. References [1] K.L. Hobbs, P. Larson, G. Lian, J. Keay, M. Johnson, Nano-Lett. 4 (2004) 167.
2185
[2] Z.K. Wang, H. Lim, H. Liu, S. Ng, M. Kuok, L. Tay, D. Lock wood, M. Cottam, K. Hobbs, P. Larson, J. Keay, G. Lian, M. Johnson, Phys. Rev. Lett. 94 (2005) 137208. [3] H. Leblond, V. Veerakumar, Phys. Rev. B 70 (2004) 134413. [4] T.M. Sharon, A. Maradudin, J. Phys. Chem. Solids 38 (1977) 977. [5] T. Wolfram, R. De Wames, Prog. Surf. Sci. 2 (1972) 233. [6] N.S. Almeida, G. Farias, N. Oliveira, E. Vasconcelos, Phys. Rev. B 48 (1993) 9839. [7] F. Schott, T. Tao, R. Freibun, J. Appl. Phys. 38 (1967) 3015.